Problem 12 - Algebra and Trigonometry

Question

\(3-24=A\) pair of parametric equations is given. (a) Sketch the curve represented by the parametric equations. (b) Find a rectangular coordinate equation for the curve by eliminating the parameter. $$ x=|t|, \quad y=|1-| t| | $$

Step-by-Step Solution

Answer

(a) Sketch results in a 'V' shape centered at (1,0). (b) Rectangular equation is \(y = |1 - x|\).

Step 1: Analyze the Parametric Equations

The parametric equations given are \(x = |t|\) and \(y = |1 - |t||\). The parameter \(t\) can be any real number, which means \(|t|\) takes non-negative values. For each value of \(|t|\), you compute \(x\) and \(y\).

Step 2: Determine the Relationship between x and t

Since \(x = |t|\), given any \(x\), \(x\) is non-negative. \(t\) must satisfy \(t = x\) or \(t = -x\), and thus, \(|t| = x\). Substitute \(x\) for \(|t|\) in the next parametric equation for \(y\).

Step 3: Substitute to Eliminate the Parameter

Substitute \(|t| = x\) into the second parametric equation: \(y = |1 - |t|| = |1 - x|\).

Step 4: Rectangular Coordinate Equation

The rectangular coordinate equation obtained is \(y = |1 - x|\). This equation describes the curve without the parameter \(t\). This is the graph of the absolute value function, which forms a 'V' shape centered at \(x = 1\).

Step 5: Sketch the Curve

To sketch the curve, note that when \(x < 1\), \(y = 1-x\), and when \(x > 1\), \(y = x-1\). At \(x = 1\), \(y = 0\). This graph makes a 'V' with the vertex at \((1,0)\) opening upwards. The line segments are \(y = 1-x\) for \(x \leq 1\) and \(y = x-1\) for \(x \geq 1\).

Key Concepts

Rectangular Coordinate Equation

In mathematics, converting a set of parametric equations into a rectangular coordinate equation can simplify understanding and graphing. When you have two equations like \(x = |t|\) and \(y = |1 - |t||\), they both depend on the parameter \(t\). By eliminating this parameter, a single, unified equation in terms of \(x\) and \(y\) is achieved.To eliminate \(t\), notice that from \(x = |t|\), \(t\) can equal \(x\) or \(-x\), leading to \(|t| = x\). Substitute into the second parametric equation: \(y = |1 - x|\). The rectangular equation \(y = |1 - x|\) beautifully simplifies the parametric form. This type of conversion helps reveal the shape and behavior of the curve without relying on the parametric variable.Rectangular coordinate equations are invaluable for graph interpretation and allow you to easily apply algebraic methods for analysis.

Absolute Value Function

The absolute value function, noted as \(|x|\), measures the distance of any real number \(x\) from zero. It only outputs non-negative values, so regardless of whether \(x\) is positive or negative, \(|x|\) will be positive.Applied to equations, this function often results in a 'V' shaped graph when plotted on a coordinate plane. Observing \(y = |1 - x|\), you can recognize it as an absolute value function. It's centered at \(x = 1\), transforming regular linear pieces into a continuous, angular shape. To examine the structure:

• For \(x < 1\), \(y = 1 - x\), creating a descending line from the left.
• For \(x > 1\), the equation adjusts to \(y = x - 1\), an ascending line from the right.

Understanding how these linear segments combine, mediated by the absolute value, provides insights into graph symmetry and behavior, crucial for sketching and analyzing the curve.

Graph Sketching

Graph sketching involves visualizing the behavior of functions and equations on paper. It helps you interpret mathematical equations in a visual form. For \(y = |1 - x|\), you should recognize the 'V' shape characteristic of absolute value functions.Here's how you can sketch this particular curve:1. **Identify Key Points**: Start by noting where the equation changes direction, which is at \(x = 1\). Here, \(y = 0\) marks the vertex of the 'V'. 2. **Plot Linear Segments**: From the vertex, draw the two respective line segments: - For \(x \leq 1\), plot \(y = 1 - x\), which descends towards the vertex. - For \(x \geq 1\), plot \(y = x - 1\), ascending from the vertex. 3. **Ensure Continuity**: Make sure the lines meet smoothly at \((1,0)\), with no breaks.Graph sketching not only solidifies understanding but also optimizes one's ability to interpret data visually. With practice, reading and generating these visual representations becomes more intuitive.